Hyperbolas are conic sections formed when a plane intersects a pair of cones. Hyperbolas have the characteristic that the difference of the distances from any point on the curve to the two foci is equal to a constant. Hyperbolas are made up of two branches that have a parabolic shape. All hyperbolas have two lines of symmetry, which intersect at the center.

Here, we will look at a more detailed definition of hyperbolas and we will learn about some of their most important characteristics.

##### PRECALCULUS

**Relevant for**…

Learning about the fundamental characteristics of hyperbolas.

##### PRECALCULUS

**Relevant for**…

Learning about the fundamental characteristics of hyperbolas.

## Definition of a hyperbola

A hyperbola is defined as the set of points so that the difference of the distances to the two foci is a constant. Hyperbolas are made up of two branches, which are a reflection of each other. Each branch of the hyperbola is similar to a parabola and has a focus and a vertex.

Hyperbolas are also defined as conic sections that are obtained at the intersection of a plane with a pair of cones. The plane cuts both bases of the cones at a certain angle.

## Main characteristics of a hyperbola

The main characteristics of a hyperbola are:

- Hyperbolas have two focal points, called foci.
- The eccentricity of the hyperbolas is greater than 1.
- The difference of each distance from a point on the hyperbola to the two foci is constant.
- Hyperbolas have two axes of symmetry, one axis passes through the foci and the other axis is perpendicular to the first.
- The intersection of the lines of symmetry is the center of the hyperbola.
- Hyperbolas have two asymptotes, towards which they approach, but never touch.
- The asymptotes also intersect at the center of the hyperbola.

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## Equation of a hyperbola

The form of the hyperbola equation depends on whether the hyperbola is centered at the origin or centered outside the origin.

When we have a hyperbola centered on the origin, its general equation is:

$latex \frac{{{x}^2}}{{{a}^2}}-\frac{{{y}^2}}{{{b}^2}}=1$ |

where *a* represents the length of the segment that extends between the two vertices of the hyperbola and *b* is found with the equation $latex {{b}^2}={{a}^2}({{e}^2}-1)$, where, *e* is the eccentricity.

If the center of the hyperbola is located outside the origin, the equation of the hyperbola is:

$latex \frac{{{(x-h)}^2}}{{{a}^2}}-\frac{{{(y-k)}^2}}{{{b}^2}}=1$ |

where, $latex (h, k)$ are the coordinates of the center of the hyperbola.

## See also

Interested in learning more about hyperbolas? Take a look at these pages:

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